PARAMETRIC EQUATIONS
Curve defined by; x = f(t), y g(t), t = parameter
Cartesian equation obtained when t is eliminated
if x = f(θ), y = g(θ): use
sinθ + cosθ = 1, cos2θ = cos²θ - 1 = 1 - 2sin²θ, sin2θ = 2sinθcosθ


VECTORS
a = b |a| = |b|, a & b same direction
a = - b |a| = |b|, a & b opposite direction
a = unit vector |a| = 1
a = zero vector |a| = 0 a + b = b + a (commutative law)
unit vector in direction of a:


Parallel vectors b = ka; k < 0: opposite direction, k > 0: same direction |b| = |k||a|

Triangle law:


Parallelogram law:




pa + qb = ra + sb p = r, q = s (a & b = non-parallel and non-zero vectors)

Position vectors
(1) relative to origin O, position vector of P is , (2)A, B and C are collinear pts

Vectors in a Cartesian plane
P = (x, y), = xi + yj (i = unit vector along x-axis, j = unit vector along y-axis)

unit vector in direction of

      =

SCALAR PRODUCT
of two non-zero vectors a . b = |a||b|cosθ (θ = angle bet a and b)
a . b = b . a (commutative law)
a . (b + c) = a . b + a . c (distributive law)
λ(a . b) = a . (λb) = (λa) . b = l |a||b|cosθ
a . b = 0 a & b are perpendicular vectors
a . a = |a|² = a²


of vectors in the Cartesian plane
a = pi + qj, b = ri + sj

a . b = pr + qs
a is perpendicular to b a . b = 0 pr + qs = 0
a . a = p² + q²


Amaths1
Amaths2
Amaths3
Amaths4

Back to 'O' level notes index